Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,23)(17,28)(22,38)(24,33)(25,32)(27,37)(29,36)(30,35)(31,40)(34,39);(1,3,6,4)(2,7,5,8)(9,11,15,25)(10,16,20,33)(12,27,30,39)(13,24,14,23)(17,34,36,37)(18,21,19,32)(22,29,40,28)(26,38,35,31);(1,4,6,3)(2,7,5,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,36)(21,37)(23,27)(24,39)(25,40)(32,34)(33,38);(1,5,6,2)(3,8,4,7)(10,18)(12,28)(13,14)(16,21)(19,20)(22,27)(23,24)(26,35)(29,30)(31,38)(32,33)(39,40)Polycyclic presentation:
2^9|a^2=ef,b^2=efi,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367342);